Problem 209 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(\alpha, \beta\) be real numbers and \(z\) a complex number. If \(z^{2}+\alpha z+\beta=0\) has two distinct roots on the line \(\operatorname{Re}(z)=1\), then it is necessary that (A) \(\beta \in(-1,0)\) (B) \(|\beta|=1\) (C) \(\beta \in(1, \infty)\) (D) \(\beta \in(0,1)\)

Step-by-Step Solution

Answer

The correct option is (C) \(\beta \in (1, \infty)\).

Step 1: Define the Roots

Since the roots of the quadratic equation lie on the line \(\operatorname{Re}(z) = 1\), we can represent them as \(z_1 = 1 + ai\) and \(z_2 = 1 + bi\) for some real numbers \(a\) and \(b\). Furthermore, since they are distinct, \(a eq b\).

Step 2: Use the Sum of Roots Formula

According to Vieta's formulas, the sum of the roots, \(z_1 + z_2\), is equal to \(-\alpha\). That is, \((1 + ai) + (1 + bi) = 2 + (a+b)i = -\alpha\). This implies \(\alpha = -2\) and \(a + b = 0\).

Step 3: Use the Product of Roots Formula

Vieta's formulas also tell us that the product of the roots, \(z_1 z_2\), equals \(\beta\). This means:\[(1 + ai)(1 + bi) = \beta\]Expanding this, we get \(1 + (a+b)i - ab = \beta\). Since \(a + b = 0\), this simplifies to \(1 - ab = \beta\).

Step 4: Analyze the Value of \(\beta\)

From the product formula \(1 - ab = \beta\), we need \(\beta\) to be positive because if \(ab\) is negative, then \(1 - ab\) becomes greater than 1. Since \(aeq b\), \(ab\) is negative (one positive and one negative). Thus, this leads to \(\beta\) being greater than 1.

Step 5: Verify the Condition

Since \(ab\) is negative (as \(aeq b\)), the expression \(1 - ab > 1\) implies \(\beta > 1\). Thus, \(\beta \in (1, \infty)\).

Key Concepts

Quadratic Equations

A quadratic equation is a mathematical expression of the form \(ax^2 + bx + c = 0\). It features a variable (typically \(x\)), and its highest power is the square (\(x^2\)).
The coefficients \(a\), \(b\), and \(c\) are usually real or complex numbers. In our exercise, the quadratic equation is \(z^2 + \alpha z + \beta = 0\), where \(z\) is a complex number.
Quadratic equations can have different types of roots, which are the solutions to the equation. Roots could be real, complex, or repeated.
In this example, the roots are complex and lie on the line \(\operatorname{Re}(z) = 1\). This means the real part of each root is exactly 1.
Keep in mind that quadratic equations always have two roots, which could be equal or distinct.
When the roots are distinct, like in our exercise, they have different imaginary parts.

Vieta's Formulas

Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots, providing an efficient way to connect them without fully solving the equation.
For a quadratic equation \(ax^2 + bx + c = 0\), the roots \(x_1\) and \(x_2\) are connected to the coefficients through these relationships:

• Sum of roots: \(x_1 + x_2 = -\frac{b}{a}\)
• Product of roots: \(x_1 x_2 = \frac{c}{a}\)

In our exercise, \(z_1\) and \(z_2\) are the roots of the given equation \(z^2 + \alpha z + \beta = 0\).
Applying Vieta's formulas, we find:

• \(z_1 + z_2 = -\alpha = 2\)
• \(z_1 z_2 = \beta\)

These relationships help us understand how \(\alpha\) and \(\beta\) influence the roots.
By knowing the real parts of \(z_1\) and \(z_2\) are both 1, it simplifies finding solutions by considering only the imaginary parts.

Line in the Complex Plane

The complex number plane, also known as the Argand plane, is a representation where the x-axis represents the real part and the y-axis represents the imaginary part of complex numbers.
A line in this plane is often defined by a constant value either in the real or imaginary component.
In the exercise, roots lie on the line \(\operatorname{Re}(z) = 1\). This means each root can be expressed as \(1 + yi\) where \(y\) is a real number.
With complex numbers being represented in the form \(a + bi\), any line parallel to the imaginary axis is defined by the real parts being constant.
Understanding the position of roots on this line helps solve the problem because it determines the equations you can use.
With \(z_1 = 1 + ai\) and \(z_2 = 1 + bi\), their distinctness is underlined by different \(a\) and \(b\) values while still aligning with Vieta's formulas. This affects calculations of properties like the product of roots \(1 - ab = \beta\), steering us to conclude conditions like \(\beta > 1\).
Such geometric interpretations in the complex plane make solving quadratic equations more intuitive.